suppose for k<n it is true
by induction |a1+...an-1|<=|a1|+...|an-1|
I'm just trying to learn induction from scratch and my book asks me to proof the generalized version of the triangle inequality.
If are arbitrary real numbers, then
My solution is as follows:
Let be the preposition that
and are true. is the normal triangle inequality, so i assume i don't need to show that it is true.
With the induction hypothesis i assume that is true.
We are trying to show that
this is true because according to the induction hypothesis. Also according to the transitive property of real numbers if a< b, then a+c < b+c.
Since is true then
Therefore it is true that
Is my proof correct ?
And is there a better way of doing this?
If and then
Let ofcourse and are true.
According to the induction hypothesis, we assume is true.
Okay moving on ...
We have to show this is true and
Doing some algebra we get
which is true because and and
Is this proof correct and is there a more clever way of doing this É
Thanks a lot this is a more clever way of doing it. Thanks again.
But is my way correct É ( My keyboard is acting up)